What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?

Question

What is the simplest $\Bbb{R}\to\Bbb{R}$ function with two peaks and a valley?

I have a set of points in $\Bbb{R^2}$ and I would like to fit a curve to the points, the points approximately lie on a curve like the one depicted in the following figure:

My points are such that $a$ (the width of the valley) is almost constant while $b$ (the height of the peaks with respect to the valley) can change.

My informal definition of "simplest" is based on the following requirements:

1. the function should be sufficient smooth
2. the fit should be easy to do with some off-the-shelf algorithm
3. I have just the points lying on the curve in figure, so I think that the function should smoothly go to zero to the left of the left peak and to the right of the right peak.

My goal is to estimate $b$.

Answer 1

When you say that you "think that the function should smoothly go to zero to the left of the left peak and to the right of the right peak", notice that it does not have to. Moreover, both $a$ and $b$ can be chosen arbitrarily.

The easiest way to do this would be to take a polynomial. If you want the simplest polynomial with two maxima at $(\pm\!\tfrac{1}{2}a,b)$ and one minimum at $(0,0)$, then $$\operatorname{f}(x) = \frac{8b}{a^4}(a^2-2x^2)x^2$$

If you insist the function to tend to zero then you need to edit your question.

Answer 2

If you want it to go to zero at $\pm \infty$, how about $(dx^2+f)\exp(-cx^2)$? Very smooth, easy to calculate, shaped like you want.